path: root/src/scalar_impl.h

/**********************************************************************
 * Copyright (c) 2014 Pieter Wuille                                   *
 * Distributed under the MIT software license, see the accompanying   *
 * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
 **********************************************************************/

#ifndef _SECP256K1_SCALAR_IMPL_H_
#define _SECP256K1_SCALAR_IMPL_H_

#include <string.h>

#include "group.h"
#include "scalar.h"

#if defined HAVE_CONFIG_H
#include "libsecp256k1-config.h"
#endif

#if defined(USE_SCALAR_4X64)
#include "scalar_4x64_impl.h"
#elif defined(USE_SCALAR_8X32)
#include "scalar_8x32_impl.h"
#else
#error "Please select scalar implementation"
#endif

typedef struct {
#ifndef USE_NUM_NONE
    secp256k1_num_t order;
#endif
#ifdef USE_ENDOMORPHISM
    secp256k1_scalar_t minus_lambda, minus_b1, minus_b2, g1, g2;
#endif
} secp256k1_scalar_consts_t;

static const secp256k1_scalar_consts_t *secp256k1_scalar_consts = NULL;

static void secp256k1_scalar_start(void) {
    if (secp256k1_scalar_consts != NULL)
        return;

    /* Allocate. */
    secp256k1_scalar_consts_t *ret = (secp256k1_scalar_consts_t*)malloc(sizeof(secp256k1_scalar_consts_t));

#ifndef USE_NUM_NONE
    static const unsigned char secp256k1_scalar_consts_order[] = {
        0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
        0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,
        0xBA,0xAE,0xDC,0xE6,0xAF,0x48,0xA0,0x3B,
        0xBF,0xD2,0x5E,0x8C,0xD0,0x36,0x41,0x41
    };
    secp256k1_num_set_bin(&ret->order, secp256k1_scalar_consts_order, sizeof(secp256k1_scalar_consts_order));
#endif
#ifdef USE_ENDOMORPHISM
    /**
     * Lambda is a scalar which has the property for secp256k1 that point multiplication by
     * it is efficiently computable (see secp256k1_gej_mul_lambda). */
    static const unsigned char secp256k1_scalar_consts_lambda[32] = {
         0x53,0x63,0xad,0x4c,0xc0,0x5c,0x30,0xe0,
         0xa5,0x26,0x1c,0x02,0x88,0x12,0x64,0x5a,
         0x12,0x2e,0x22,0xea,0x20,0x81,0x66,0x78,
         0xdf,0x02,0x96,0x7c,0x1b,0x23,0xbd,0x72
    };
    /**
     * "Guide to Elliptic Curve Cryptography" (Hankerson, Menezes, Vanstone) gives an algorithm
     * (algorithm 3.74) to find k1 and k2 given k, such that k1 + k2 * lambda == k mod n, and k1
     * and k2 have a small size.
     * It relies on constants a1, b1, a2, b2. These constants for the value of lambda above are:
     *
     * - a1 =      {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15}
     * - b1 =     -{0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28,0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3}
     * - a2 = {0x01,0x14,0xca,0x50,0xf7,0xa8,0xe2,0xf3,0xf6,0x57,0xc1,0x10,0x8d,0x9d,0x44,0xcf,0xd8}
     * - b2 =      {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15}
     *
     * The algorithm then computes c1 = round(b1 * k / n) and c2 = round(b2 * k / n), and gives
     * k1 = k - (c1*a1 + c2*a2) and k2 = -(c1*b1 + c2*b2). Instead, we use modular arithmetic, and
     * compute k1 as k - k2 * lambda, avoiding the need for constants a1 and a2.
     *
     * g1, g2 are precomputed constants used to replace division with a rounded multiplication
     * when decomposing the scalar for an endomorphism-based point multiplication.
     *
     * The possibility of using precomputed estimates is mentioned in "Guide to Elliptic Curve
     * Cryptography" (Hankerson, Menezes, Vanstone) in section 3.5.
     *
     * The derivation is described in the paper "Efficient Software Implementation of Public-Key
     * Cryptography on Sensor Networks Using the MSP430X Microcontroller" (Gouvea, Oliveira, Lopez),
     * Section 4.3 (here we use a somewhat higher-precision estimate):
     * d = a1*b2 - b1*a2
     * g1 = round((2^272)*b2/d)
     * g2 = round((2^272)*b1/d)
     *
     * (Note that 'd' is also equal to the curve order here because [a1,b1] and [a2,b2] are found
     * as outputs of the Extended Euclidean Algorithm on inputs 'order' and 'lambda').
     */
    static const unsigned char secp256k1_scalar_consts_minus_b1[32] = {
        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
        0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28,
        0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3
    };
    static const unsigned char secp256k1_scalar_consts_b2[32] = {
        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
        0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,
        0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15
    };
    static const unsigned char secp256k1_scalar_consts_g1[32] = {
        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
        0x00,0x00,0x00,0x00,0x00,0x00,0x30,0x86,
        0xd2,0x21,0xa7,0xd4,0x6b,0xcd,0xe8,0x6c,
        0x90,0xe4,0x92,0x84,0xeb,0x15,0x3d,0xab
    };
    static const unsigned char secp256k1_scalar_consts_g2[32] = {
        0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,
        0x00,0x00,0x00,0x00,0x00,0x00,0xe4,0x43,
        0x7e,0xd6,0x01,0x0e,0x88,0x28,0x6f,0x54,
        0x7f,0xa9,0x0a,0xbf,0xe4,0xc4,0x22,0x12
    };

    secp256k1_scalar_set_b32(&ret->minus_lambda, secp256k1_scalar_consts_lambda, NULL);
    secp256k1_scalar_negate(&ret->minus_lambda, &ret->minus_lambda);
    secp256k1_scalar_set_b32(&ret->minus_b1, secp256k1_scalar_consts_minus_b1, NULL);
    secp256k1_scalar_set_b32(&ret->minus_b2, secp256k1_scalar_consts_b2, NULL);
    secp256k1_scalar_negate(&ret->minus_b2, &ret->minus_b2);
    secp256k1_scalar_set_b32(&ret->g1, secp256k1_scalar_consts_g1, NULL);
    secp256k1_scalar_set_b32(&ret->g2, secp256k1_scalar_consts_g2, NULL);
#endif

    /* Set the global pointer. */
    secp256k1_scalar_consts = ret;
}

static void secp256k1_scalar_stop(void) {
    if (secp256k1_scalar_consts == NULL)
        return;

    secp256k1_scalar_consts_t *c = (secp256k1_scalar_consts_t*)secp256k1_scalar_consts;
    secp256k1_scalar_consts = NULL;
    free(c);
}

#ifndef USE_NUM_NONE
static void secp256k1_scalar_get_num(secp256k1_num_t *r, const secp256k1_scalar_t *a) {
    unsigned char c[32];
    secp256k1_scalar_get_b32(c, a);
    secp256k1_num_set_bin(r, c, 32);
}

static void secp256k1_scalar_order_get_num(secp256k1_num_t *r) {
    *r = secp256k1_scalar_consts->order;
}
#endif

static void secp256k1_scalar_inverse(secp256k1_scalar_t *r, const secp256k1_scalar_t *x) {
    /* First compute x ^ (2^N - 1) for some values of N. */
    secp256k1_scalar_t x2, x3, x4, x6, x7, x8, x15, x30, x60, x120, x127;

    secp256k1_scalar_sqr(&x2,  x);
    secp256k1_scalar_mul(&x2, &x2,  x);

    secp256k1_scalar_sqr(&x3, &x2);
    secp256k1_scalar_mul(&x3, &x3,  x);

    secp256k1_scalar_sqr(&x4, &x3);
    secp256k1_scalar_mul(&x4, &x4,  x);

    secp256k1_scalar_sqr(&x6, &x4);
    secp256k1_scalar_sqr(&x6, &x6);
    secp256k1_scalar_mul(&x6, &x6, &x2);

    secp256k1_scalar_sqr(&x7, &x6);
    secp256k1_scalar_mul(&x7, &x7,  x);

    secp256k1_scalar_sqr(&x8, &x7);
    secp256k1_scalar_mul(&x8, &x8,  x);

    secp256k1_scalar_sqr(&x15, &x8);
    for (int i=0; i<6; i++)
        secp256k1_scalar_sqr(&x15, &x15);
    secp256k1_scalar_mul(&x15, &x15, &x7);

    secp256k1_scalar_sqr(&x30, &x15);
    for (int i=0; i<14; i++)
        secp256k1_scalar_sqr(&x30, &x30);
    secp256k1_scalar_mul(&x30, &x30, &x15);

    secp256k1_scalar_sqr(&x60, &x30);
    for (int i=0; i<29; i++)
        secp256k1_scalar_sqr(&x60, &x60);
    secp256k1_scalar_mul(&x60, &x60, &x30);

    secp256k1_scalar_sqr(&x120, &x60);
    for (int i=0; i<59; i++)
        secp256k1_scalar_sqr(&x120, &x120);
    secp256k1_scalar_mul(&x120, &x120, &x60);

    secp256k1_scalar_sqr(&x127, &x120);
    for (int i=0; i<6; i++)
        secp256k1_scalar_sqr(&x127, &x127);
    secp256k1_scalar_mul(&x127, &x127, &x7);

    /* Then accumulate the final result (t starts at x127). */
    secp256k1_scalar_t *t = &x127;
    for (int i=0; i<2; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<4; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x3); /* 111 */
    for (int i=0; i<2; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<2; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<2; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<4; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x3); /* 111 */
    for (int i=0; i<3; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x2); /* 11 */
    for (int i=0; i<4; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x3); /* 111 */
    for (int i=0; i<5; i++) /* 00 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x3); /* 111 */
    for (int i=0; i<4; i++) /* 00 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x2); /* 11 */
    for (int i=0; i<2; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<2; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<5; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x4); /* 1111 */
    for (int i=0; i<2; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<3; i++) /* 00 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<4; i++) /* 000 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<2; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<10; i++) /* 0000000 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x3); /* 111 */
    for (int i=0; i<4; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x3); /* 111 */
    for (int i=0; i<9; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x8); /* 11111111 */
    for (int i=0; i<2; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<3; i++) /* 00 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<3; i++) /* 00 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<5; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x4); /* 1111 */
    for (int i=0; i<2; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<5; i++) /* 000 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x2); /* 11 */
    for (int i=0; i<4; i++) /* 00 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x2); /* 11 */
    for (int i=0; i<2; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<8; i++) /* 000000 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x2); /* 11 */
    for (int i=0; i<3; i++) /* 0 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, &x2); /* 11 */
    for (int i=0; i<3; i++) /* 00 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<6; i++) /* 00000 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(t, t, x); /* 1 */
    for (int i=0; i<8; i++) /* 00 */
        secp256k1_scalar_sqr(t, t);
    secp256k1_scalar_mul(r, t, &x6); /* 111111 */
}

static void secp256k1_scalar_inverse_var(secp256k1_scalar_t *r, const secp256k1_scalar_t *x) {
#if defined(USE_SCALAR_INV_BUILTIN)
    secp256k1_scalar_inverse(r, x);
#elif defined(USE_SCALAR_INV_NUM)
    unsigned char b[32];
    secp256k1_scalar_get_b32(b, x);
    secp256k1_num_t n;
    secp256k1_num_set_bin(&n, b, 32);
    secp256k1_num_mod_inverse(&n, &n, &secp256k1_scalar_consts->order);
    secp256k1_num_get_bin(b, 32, &n);
    secp256k1_scalar_set_b32(r, b, NULL);
#else
#error "Please select scalar inverse implementation"
#endif
}

#ifdef USE_ENDOMORPHISM
static void secp256k1_scalar_split_lambda_var(secp256k1_scalar_t *r1, secp256k1_scalar_t *r2, const secp256k1_scalar_t *a) {
    VERIFY_CHECK(r1 != a);
    VERIFY_CHECK(r2 != a);
    secp256k1_scalar_t c1, c2;
    secp256k1_scalar_mul_shift_var(&c1, a, &secp256k1_scalar_consts->g1, 272);
    secp256k1_scalar_mul_shift_var(&c2, a, &secp256k1_scalar_consts->g2, 272);
    secp256k1_scalar_mul(&c1, &c1, &secp256k1_scalar_consts->minus_b1);
    secp256k1_scalar_mul(&c2, &c2, &secp256k1_scalar_consts->minus_b2);
    secp256k1_scalar_add(r2, &c1, &c2);
    secp256k1_scalar_mul(r1, r2, &secp256k1_scalar_consts->minus_lambda);
    secp256k1_scalar_add(r1, r1, a);
}
#endif

#endif